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Cayley diagram

C2×#13272 #495346

order512
nilpotency class4
isoclinismisoclinic class of #2230
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4

Aliases

direct productC2×#13272

Subgroups

subgroupG/H
center, socleC23#476
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC24AGL(1,ℤ/8)
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

962 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 4 generators on 22 points:

(1,2);(4,6)(7,8,10,16,12,20,11,15,14,17,21,22,13,9,18,19);(3,4,5,6)(7,9,13,8,14,20,12,17)(10,16,18,15,21,22,11,19);(7,10,12,11,14,21,13,18)(8,15,9,16,17,19,20,22)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#512418, C2×#512192, C2×#495541
maximal normal subgroup ofC2×#512418, C2×#512192, C2×#495541

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402542);